Editing Laplace transform (section)

== Table of selected Laplace transforms ==
{{main article|List of Laplace transforms}}

The following table provides Laplace transforms for many common functions of a single variable.<ref>{{Citation |edition=3rd |page=455 |first1=K. F. |last1=Riley |first2=M. P. |last2=Hobson |first3=S. J. |last3=Bence |title=Mathematical methods for physics and engineering |publisher=Cambridge University Press |year=2010 |isbn=978-0-521-86153-3}}</ref><ref>{{Citation |first1=J. J. |last1=Distefano |first2=A. R. |last2=Stubberud |first3=I. J. |last3=Williams |page=78 |title=Feedback systems and control |edition=2nd |publisher=McGraw-Hill |series=Schaum's outlines |year=1995 |isbn=978-0-07-017052-0}}</ref> For definitions and explanations, see the ''Explanatory Notes'' at the end of the table.

Because the Laplace transform is a linear operator,
* The Laplace transform of a sum is the sum of Laplace transforms of each term.<!--
--><math display=block>\mathcal{L}\{f(t) + g(t)\}  = \mathcal{L}\{f(t)\} + \mathcal{L}\{ g(t)\}  </math>
* The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.<!--
--><math display=block>\mathcal{L}\{a f(t)\}  = a \mathcal{L}\{ f(t)\}</math>

Using this linearity, and various [[List of trigonometric identities|trigonometric]], [[Hyperbolic function|hyperbolic]], and complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others more quickly than by using the definition directly.

The unilateral Laplace transform takes as input a function whose time domain is the [[non-negative]] reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, {{math|''u''(''t'')}}.

The entries of the table that involve a time delay {{math|''τ''}} are required to be [[causal system|causal]] (meaning that {{math|''τ'' > 0}}).  A causal system is a system where the [[impulse response]] {{math|''h''(''t'')}} is zero for all time {{mvar|t}} prior to {{math|1=''t'' = 0}}. In general, the region of convergence for causal systems is not the same as that of [[anticausal system]]s.

{| class="wikitable" style="text-align: center;"
|+ Selected Laplace transforms
|-
 ! scope="col" | Function
 ! scope="col" | Time domain <br> <math>f(t) = \mathcal{L}^{-1}\{F(s)\}</math> 
 ! scope="col" | Laplace {{math|s}}-domain <br/> <math>F(s) = \mathcal{L}\{f(t)\}</math> 
 ! scope="col" | Region of convergence 
 ! scope="col" | Reference
|-
 ! scope="row" | unit impulse
 | <math> \delta(t) \ </math> 
 | <math> 1  </math> 
 | all {{math|''s''}}
 | inspection
|-
 ! scope="row" | delayed impulse 
 | <math> \delta(t - \tau) \ </math> 
 | <math> e^{-\tau s} \ </math> 
 | all {{math|''s''}} 
 | time shift of<br>unit impulse
|-
 ! scope="row"| unit step
 | <math> u(t) \ </math> 
 | <math> { 1 \over s } </math> 
 | <math> \operatorname{Re}(s) > 0 </math> 
 | integrate unit impulse
|-
 ! scope="row" | delayed unit step 
 | <math> u(t - \tau) \ </math> 
 | <math> \frac 1 s e^{-\tau s} </math> 
 | <math> \operatorname{Re}(s) > 0 </math> 
 | time shift of<br>unit step
|-
 ! scope="row" | product of delayed function and delayed step
 | <math> f(t-\tau)u(t-\tau) </math>
 | <math> e^{-s\tau}\mathcal{L}\{f(t)\}</math>
 |
 | u-substitution, <math>u=t-\tau</math>
|-
 !rectangular impulse
 | <math> u (t) - u(t - \tau) </math>
 | <math> \frac{1}{s}(1 - e^{-\tau s}) </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 |
|-
 ! scope="row" | [[ramp function|ramp]]
 | <math> t \cdot u(t)\ </math>
 | <math>\frac 1 {s^2}</math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | integrate unit<br>impulse twice
|-
 ! scope="row" | {{math|''n''}}th power <br/> (for integer {{math|''n''}}) 
 | <math> t^n \cdot u(t) </math>
 | <math> { n! \over s^{n + 1} } </math>
 | <math> \operatorname{Re}(s) > 0 </math> <br/> ({{math|''n'' > −1}})
 | integrate unit<br>step {{math|''n''}} times
|-
 ! scope="row" | {{math|''q''}}th power <br /> (for complex {{math|''q''}}) 
 | <math> t^q \cdot u(t) </math>
 | <math> { \operatorname{\Gamma}(q + 1) \over s^{q + 1} } </math>
 |  <math> \operatorname{Re}(s) > 0 </math>  <br/>  <math> \operatorname{Re}(q) > -1 </math> 
 | <ref>{{cite book |title=Mathematical Handbook of Formulas and Tables |edition=3rd |first1=S. |last1=Lipschutz |first2=M. R. |last2=Spiegel |first3=J. |last3=Liu |series=Schaum's Outline Series |publisher=McGraw-Hill |page=183 |year=2009 |isbn=978-0-07-154855-7}} – provides the case for real {{math|''q''}}.</ref><ref>http://mathworld.wolfram.com/LaplaceTransform.html – Wolfram Mathword provides case for complex {{math|''q''}}</ref>
|-
 ! scope="row" | {{math|''n''}}th root 
 | <math> \sqrt[n]{t} \cdot u(t) </math>
 | <math> { 1 \over s^{\frac 1 n + 1} } \operatorname{\Gamma}\left(\frac 1 n + 1\right) </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | Set {{math|''q'' {{=}} 1/''n''}} above.
|-
 ! scope="row" | {{math|''n''}}th power with frequency shift 
 | <math>t^{n} e^{-\alpha t} \cdot u(t) </math>
 | <math>\frac{n!}{(s+\alpha)^{n+1}} </math>
 | <math> \operatorname{Re}(s) > -\alpha </math> 
 | Integrate unit step,<br/>apply frequency shift
|-
 ! scope="row" | delayed {{math|''n''}}th power <br /> with frequency shift 
 | <math>(t-\tau)^n e^{-\alpha (t-\tau)} \cdot u(t-\tau) </math>
 | <math> \frac{n! \cdot e^{-\tau s}}{(s+\alpha)^{n+1}} </math>
 |  <math> \operatorname{Re}(s) > -\alpha </math> 
 | integrate unit step,<br>apply frequency shift,<br>apply time shift
|-
 ! scope="row" | [[exponential decay]]
 | <math> e^{-\alpha t} \cdot u(t)   </math>
 | <math> { 1 \over s+\alpha } </math>
 | <math> \operatorname{Re}(s) > -\alpha </math> 
 | Frequency shift of<br>unit step
|-
 ! scope="row" | [[Two-sided Laplace transform|two-sided]] exponential decay <br>(only for bilateral transform)
 | <math> e^{-\alpha|t|}  \ </math>
 | <math> { 2\alpha \over \alpha^2 - s^2 } </math>
 | <math>  -\alpha < \operatorname{Re}(s) < \alpha </math> 
 | Frequency shift of<br>unit step
|-
 ! scope="row" | exponential approach 
 | <math>(1-e^{-\alpha t})  \cdot u(t)  \ </math>
 | <math>\frac{\alpha}{s(s+\alpha)} </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | unit step minus<br/>exponential decay
|-
 ! scope="row" | [[sine]]
 | <math> \sin(\omega t) \cdot u(t) \ </math>
 | <math> { \omega \over s^2 + \omega^2  } </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | {{sfn|Bracewell|1978|p=227}}
|-
 ! scope="row" | [[cosine]]
 | <math> \cos(\omega t) \cdot u(t) \ </math>
 | <math> { s \over s^2 + \omega^2  } </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | {{sfn|Bracewell|1978|p=227}}
|-
 ! scope="row" | [[hyperbolic sine]]
 | <math> \sinh(\alpha t) \cdot u(t) \ </math>
 | <math> { \alpha \over s^2 - \alpha^2 } </math>
 | <math> \operatorname{Re}(s) > \left| \alpha \right| </math> 
 | {{sfn|Williams|1973|p=88}}
|-
 ! scope="row" | [[hyperbolic cosine]]
 | <math> \cosh(\alpha t) \cdot u(t) \ </math>
 | <math> { s \over s^2 - \alpha^2  } </math>
 |  <math> \operatorname{Re}(s) > \left| \alpha \right| </math> 
 | {{sfn|Williams|1973|p=88}}
|-
 ! scope="row" | exponentially decaying <br /> sine wave 
 | <math>e^{-\alpha t}  \sin(\omega t) \cdot u(t) \ </math>
 | <math> { \omega \over (s+\alpha)^2 + \omega^2  } </math>
 | <math> \operatorname{Re}(s) > - \alpha </math> 
 | {{sfn|Bracewell|1978|p=227}}
|-
 ! scope="row" | exponentially decaying <br /> cosine wave 
 | <math>e^{-\alpha t}  \cos(\omega t) \cdot u(t) \ </math>
 | <math> { s+\alpha \over (s+\alpha)^2 + \omega^2  } </math>
 | <math> \operatorname{Re}(s) > - \alpha </math> 
 | {{sfn|Bracewell|1978|p=227}}
|-
 ! scope="row" | [[natural logarithm]]
 | <math> \ln(t) \cdot u(t) </math>
 | <math> -{1 \over s} \left[ \ln(s)+\gamma \right] </math>
 |  <math> \operatorname{Re}(s) > 0 </math> 
 | {{sfn|Williams|1973|p=88}}
|-
 ! scope="row" | [[Bessel function]] <br> of the first kind, <br /> of order {{math|''n''}}
 | <math> J_n(\omega t) \cdot u(t)</math>
 | <math>\frac{ \left(\sqrt{s^2+ \omega^2}-s\right)^{\!n}}{\omega^n \sqrt{s^2 + \omega^2}}</math>
 | <math> \operatorname{Re}(s) > 0 </math> <br/> ({{math|''n'' > −1}}) 
 | {{sfn|Williams|1973|p=89}}
|-
 ! scope="row" | [[Error function]] 
 | <math> \operatorname{erf}(t) \cdot u(t) </math> 
 | <math> \frac{1}{s} e^{s^2 / 4} \!\left(1 - \operatorname{erf} \frac{s}{2} \right)</math> 
 | <math> \operatorname{Re}(s) > 0 </math>
 | {{sfn|Williams|1973|p=89}}
|-
 | colspan=5 style="text-align: left;" |'''Explanatory notes:'''
{{col-begin}}
{{col-break}}
* {{math|''u''(''t'')}} represents the [[Heaviside step function]].
* {{math|''δ''}}  represents the [[Dirac delta function]].
* {{math|Γ(''z'')}} represents the [[gamma function]].
* {{math|''γ''}} is the [[Euler&ndash;Mascheroni constant]].
{{col-break}}
*  {{math|''t''}}, a real number, typically represents ''time'', although it can represent ''any'' independent dimension.
*  {{math|''s''}} is the [[complex number|complex]] frequency domain parameter, and  {{math|Re(''s'')}} is its [[real part]].
*  {{math|''α'', ''β'', ''τ'', and ''ω''}} are [[real numbers]].
*  {{math|''n''}} is an [[integer]].
{{col-end}}
|}